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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1344.8546', 'ambient_counter': 8546, 'ambient_order': 1344, 'ambient_tex': 'D_{56}:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 56, 'characteristic': False, 'core_order': 168, 'counter': 94, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1344.8546.8.o1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.o1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '168.3', 'subgroup_hash': 3, 'subgroup_order': 168, 'subgroup_tex': 'C_3:C_{56}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8546', 'aut_centralizer_order': None, 'aut_label': '8.o1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.e1.b1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.g1.a1', '4.k1.b1', '4.o1.a1', '4.q1.a1', '4.r1.b1', '4.t1.a1', '4.v1.b1'], 'contains': ['16.d1.a1', '24.bg1.a1', '56.i1.a1'], 'core': '8.o1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [730, 336, 672, 192, 448], 'label': '1344.8546.8.o1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.o1.a1', 'normal_contained_in': ['4.g1.a1', '4.k1.b1', '4.o1.a1', '4.q1.a1', '4.r1.b1', '4.t1.a1', '4.v1.b1'], 'normal_contains': ['16.d1.a1', '56.i1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.o1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.o1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '56.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6, 6], 'aut_gens': [[1, 8], [5, 8], [1, 64], [3, 128], [57, 104]], 'aut_group': '144.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 12, 'aut_perms': [41064, 3630480, 1683, 43546344], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [7, 1, 6, 1], [8, 3, 4, 1], [12, 2, 2, 1], [14, 1, 6, 1], [21, 2, 6, 1], [28, 1, 12, 1], [42, 2, 6, 1], [56, 3, 24, 1], [84, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '24.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 1, 2], [6, 2, 1], [7, 1, 6], [8, 3, 4], [12, 2, 2], [14, 1, 6], [21, 2, 6], [28, 1, 12], [42, 2, 6], [56, 3, 24], [84, 2, 12]], 'center_label': '28.2', 'center_order': 28, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [7, 1, 6, 1], [8, 3, 4, 1], [12, 2, 2, 1], [14, 1, 6, 1], [21, 2, 6, 1], [28, 1, 12, 1], [42, 2, 6, 1], [56, 3, 24, 1], [84, 2, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 96, 'exponent': 168, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 12]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '42.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 64], [113, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 56], [2, 28]], 'label': '168.3', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C56', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 84, 'number_divisions': 14, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [7, 6], [8, 12], [12, 4], [14, 6], [21, 12], [28, 12], [42, 12], [56, 72], [84, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5, 8], [3, 8], [1, 152]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [8, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [6, 2], [12, 3], [24, 2]], 'representations': {'PC': {'code': 5440890537043499, 'gens': [1, 4], 'pres': [5, -2, -2, -2, -3, -7, 10, 26, 1283, 78]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [570407, 112901]}, 'Perm': {'d': 18, 'gens': [379406669281920, 873, 777033521510400, 1156002822374400, 403200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [56], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_{56}', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [28, 12, 4, 2, 6, 12, 14, 6], 'aut_gens': [[1, 2, 4, 8], [1, 450, 244, 232], [673, 2, 1012, 584], [1, 1122, 1252, 1337], [673, 2, 868, 1336], [1, 2, 1060, 200], [1, 898, 820, 473], [673, 898, 916, 568], [673, 674, 580, 136]], 'aut_group': None, 'aut_hash': 9161810123478319507, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 42, 'aut_perms': [583550369770708210561592191385974893571751572414179, 764084665782828898446668915131421555778631620757370, 991647396469354839691729168086794734307777193668023, 135463909557276468182970101453166955853916448883382, 145794125113320540721590920375246730756219672035544, 297947856977588547619464234385086575903881336623854, 1199688577555279090395699842565137369163176260688166, 93602433051292623675165109035862341010583153906717], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [2, 28, 1, 1], [2, 28, 2, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 3, 2, 1], [4, 6, 1, 1], [4, 28, 1, 1], [4, 84, 1, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 56, 1, 1], [6, 56, 2, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 6, 6, 1], [14, 12, 3, 1], [21, 4, 3, 1], [24, 8, 2, 1], [28, 2, 6, 1], [28, 4, 3, 1], [28, 6, 6, 1], [28, 12, 3, 1], [42, 4, 3, 1], [42, 8, 3, 1], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 1], [84, 8, 3, 1], [168, 8, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{21}.(C_6\\times D_4).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 5451351369529179689, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 2], [2, 28, 3], [2, 84, 1], [3, 2, 1], [4, 2, 2], [4, 3, 2], [4, 6, 1], [4, 28, 1], [4, 84, 3], [6, 2, 1], [6, 4, 1], [6, 56, 3], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 56, 1], [14, 2, 3], [14, 4, 3], [14, 6, 6], [14, 12, 3], [21, 4, 3], [24, 8, 2], [28, 2, 6], [28, 4, 3], [28, 6, 6], [28, 12, 3], [42, 4, 3], [42, 8, 3], [56, 4, 12], [56, 12, 12], [84, 4, 6], [84, 8, 3], [168, 8, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '672.1141', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8546, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [2, 28, 1, 3], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 3, 2, 1], [4, 6, 1, 1], [4, 28, 1, 1], [4, 84, 1, 3], [6, 2, 1, 1], [6, 4, 1, 1], [6, 56, 1, 3], [7, 2, 3, 1], [8, 4, 1, 2], [8, 12, 1, 2], [12, 4, 1, 2], [12, 56, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 6, 6, 1], [14, 12, 3, 1], [21, 4, 3, 1], [24, 8, 1, 2], [28, 2, 6, 1], [28, 4, 3, 1], [28, 6, 6, 1], [28, 12, 3, 1], [42, 4, 3, 1], [42, 8, 3, 1], [56, 4, 6, 2], [56, 12, 6, 2], [84, 4, 6, 1], [84, 8, 3, 1], [168, 8, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 29877120, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, -1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 8546, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 2, 84], 'inner_gens': [[1, 2, 4, 680], [1, 2, 4, 232], [1, 2, 4, 440], [673, 1122, 916, 8]], 'inner_hash': 1141, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_6\\times D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 60], [4, 40], [8, 7]], 'label': '1344.8546', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D56:D6', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 43, 'number_characteristic_subgroups': 78, 'number_conjugacy_classes': 123, 'number_divisions': 52, 'number_normal_subgroups': 136, 'number_subgroup_autclasses': 408, 'number_subgroup_classes': 524, 'number_subgroups': 4724, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 183], [3, 2], [4, 296], [6, 174], [7, 6], [8, 32], [12, 64], [14, 90], [21, 12], [24, 16], [28, 96], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 1], 'outer_gens': [[673, 2, 4, 8], [1, 674, 4, 8], [1, 2, 4, 1016], [1, 2, 4, 825]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [40320, 3628800, 24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 2], [6, 8], [8, 2], [12, 8], [24, 2], [48, 2]], 'representations': {'PC': {'code': 18116660533071618689973858837564429626444513796312690912598283, 'gens': [1, 2, 3, 4], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 5376, 21763, 3723, 3539, 91, 9292, 8820, 116, 22285, 5013, 141, 14350, 11670, 222, 18455]}, 'Perm': {'d': 26, 'gens': [621574835363066525758440, 355731555140686, 1621106561118, 3023250411916, 4398020226618, 5801212720938, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{56}:D_6', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 7]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
